HavenHub Math • Edition 10 • Student Workbook

Unit 4: The Triangle

"The Pythagorean Law of Squares"

The Builder's Meditation: Squaring the Foundation

A house built on a crooked corner will not stand. In the Bible, we are told that Christ is the "Chief Cornerstone" (Psalm 118:22). He is the anchor that makes the rest of the building square and true.

In this unit, you will learn the Pythagorean Theorem. This law allows you to prove that a corner is exactly 90 degrees. It connects the length of the legs to the distance of the bridge. As you square numbers and find their roots, remember that truth is the only foundation strong enough to build a life upon.

Lesson 4.1: Squares and Roots (Seed and field)

The Law of the Seed: Squaring a number ($x^2$) grows it into a field. Taking the Square Root ($\sqrt{x}$) returns the field to its original seed.
1. The Growth Drill: Calculate the field size.
$1^2 =$
$2^2 =$
$3^2 =$
$4^2 =$
$5^2 =$
$10^2 =$
$12^2 =$
$20^2 =$
2. The Redemption Drill: Find the original seed.
$\sqrt{9} =$
$\sqrt{16} =$
$\sqrt{49} =$
$\sqrt{64} =$
$\sqrt{100} =$
$\sqrt{121} =$
$\sqrt{144} =$
$\sqrt{400} =$

Lesson 4.2: Anatomy of the Corner

1. Label the Sovereign Shape.

On the triangle below:

2. The Longest Path.

Why must side $c$ be longer than side $a$ or side $b$? If you walk across a field diagonally, is it shorter or longer than walking around the edges?

Lesson 4.3 & 4.4: The Pythagorean Law

The Blueprint:
$a^2 + b^2 = c^2$

1. Square the legs. 2. Add them up. 3. Take the square root of the sum to find $c$.

1. Calculating the Bridge: Find the missing hypotenuse.
A. Leg $a=3, b=4$

$3^2 + 4^2 = c^2$

$9 + 16 = c^2$

$25 = c^2$

$c = $
B. Leg $a=5, b=12$
$c = $
C. Leg $a=8, b=6$
$c = $

Lesson 4.5: The TV Challenge (Real World Diagonals)

The Revelation: Screens are sold by their diagonal length. If you know the width and the height, you can use the Pythagorean Law to find the screen size.
1. The Home Theater: Calculate the diagonal of these screens.
A. Computer Monitor

Width ($a$): 12 in.

Height ($b$): 9 in.

Diagonal ($c$):
B. Big Screen TV

Width ($a$): 40 in.

Height ($b$): 30 in.

Diagonal ($c$):

The Scribe's Precision Grid: 100 Law Drills

Identify the squares and roots. Find the missing side.

1. $1^2 \rightarrow$
2. $2^2 \rightarrow$
3. $3^2 \rightarrow$
4. $4^2 \rightarrow$
5. $5^2 \rightarrow$
6. $6^2 \rightarrow$
7. $7^2 \rightarrow$
8. $8^2 \rightarrow$
9. $9^2 \rightarrow$
10. $10^2 \rightarrow$
11. $\sqrt{1} \rightarrow$
12. $\sqrt{4} \rightarrow$
13. $\sqrt{9} \rightarrow$
14. $\sqrt{16} \rightarrow$
15. $\sqrt{25} \rightarrow$
16. $\sqrt{36} \rightarrow$
17. $\sqrt{49} \rightarrow$
18. $\sqrt{64} \rightarrow$
19. $\sqrt{81} \rightarrow$
20. $\sqrt{100} \rightarrow$
21. 3, 4, c $\rightarrow$
22. 6, 8, c $\rightarrow$
23. 5, 12, c $\rightarrow$
24. 9, 12, c $\rightarrow$
25. 8, 15, c $\rightarrow$
26. $11^2 \rightarrow$ 121
27. $12^2 \rightarrow$ 144
28. $13^2 \rightarrow$ 169
29. $14^2 \rightarrow$ 196
30. $15^2 \rightarrow$ 225
31. $\sqrt{121} \rightarrow$ 11
32. $\sqrt{144} \rightarrow$ 12
33. $\sqrt{169} \rightarrow$ 13
34. $\sqrt{196} \rightarrow$ 14
35. $\sqrt{225} \rightarrow$ 15
36. $20^2 \rightarrow$ 400
37. $30^2 \rightarrow$ 900
38. $40^2 \rightarrow$ 1600
39. $50^2 \rightarrow$ 2500
40. $100^2 \rightarrow$ 10000
41. $\sqrt{400} \rightarrow$ 20
42. $\sqrt{900} \rightarrow$ 30
43. $\sqrt{1600} \rightarrow$ 40
44. $\sqrt{2500} \rightarrow$ 50
45. $\sqrt{10000} \rightarrow$ 100
46. 10, 24, c $\rightarrow$ 26
47. 12, 16, c $\rightarrow$ 20
48. 15, 20, c $\rightarrow$ 25
49. 20, 21, c $\rightarrow$ 29
50. 7, 24, c $\rightarrow$ 25
51. a=3, c=5 $\rightarrow$ b=4
52. a=6, c=10 $\rightarrow$ b=8
53. a=5, c=13 $\rightarrow$ b=12
54. a=8, c=17 $\rightarrow$ b=15
55. a=9, c=15 $\rightarrow$ b=12
56. Corner is 90 deg.
57. Triangle is Right.
58. Theorem is Law.
59. Pythagoras is name.
60. a^2+b^2=c^2.
61. Side a.
62. Side b.
63. Side c.
64. Legs.
65. Hypotenuse.
66. Root.
67. Power.
68. Square.
69. Area.
70. Length.
71. 3-4-5.
72. 5-12-13.
73. 8-15-17.
74. 7-24-25.
75. 9-40-41.
76. True Corner.
77. Square Base.
78. Plumb Line.
79. Builder's Rule.
80. Scribe's Truth.
81. Calculate.
82. Verify.
83. Measure.
84. Draw.
81. Calculate.
82. Verify.
83. Measure.
84. Draw.
85. Prove.
86. Triangle 1.
87. Triangle 2.
88. Triangle 3.
89. Law 1.0.
90. Truth 2.0.
91. $3^2=9$.
92. $4^2=16$.
93. $9+16=25$.
94. $\sqrt{25}=5$.
95. Mastered.
96. Done.
97. Square.
98. Add.
99. Root.
100. 1.00 Shalom.
101. $5^2 = 25$.
102. $12^2 = 144$.
103. $25+144 = 169$.
104. $\sqrt{169} = 13$.
105. 5-12-13.
106. $\sqrt{2} \approx 1.41$.
107. $\sqrt{3} \approx 1.73$.
108. $\sqrt{5} \approx 2.23$.
109. Estimating.
110. Rounding.
111. Legs meet.
112. Hypotenuse cuts.
113. 90 Degrees.
114. Orthogonal.
115. Perpendicular.
116. $a^2+b^2=c^2$.
117. $c = \sqrt{a^2+b^2}$.
118. $a = \sqrt{c^2-b^2}$.
119. $b = \sqrt{c^2-a^2}$.
120. One Law.

📋 The Scribe's Final Audit: Unit 4

Before you receive your Signet of Construction, you must pass the check.

  1. Can you define the Hypotenuse?
  2. Do you know the root of 144?
  3. Can you state the Pythagorean Theorem?
  4. Do you understand that the diagonal is shorter than the two legs added together?
  5. Are you ready to build square foundations in the real world?

Auditor's Signature:

The Prophet's Vision: The Law of the Builder

1. The Anchor of the Boat: A boat is anchored in a lake. The anchor line is 13 feet long. The boat is floating 12 feet away from the spot directly above the anchor. How deep is the lake?
Calculation: Depth:
2. The Ladder of Safety: A 10-foot ladder is leaning against a wall. The base of the ladder is 6 feet from the wall. How high up the wall does the ladder reach?
Calculation: Height:
3. The City Gates: A gate is 8 feet wide and 6 feet high. To keep it strong, the builder wants to put a diagonal brace across the middle. How long must the brace be?
Calculation: Brace:

The Architect's Grid: Drafting Triangles

The Task: On each grid, plot the two legs of the triangle starting from the Origin. Then draw the hypotenuse and use the Law to find its length.
1. Triangle A: Leg $a=6$ (on X-axis), Leg $b=8$ (on Y-axis).
Calculation: $c = $
2. Triangle B: Leg $a=5$ (on X-axis), Leg $b=12$ (on Y-axis).
Calculation: $c = $

The Pattern Seeker: Pythagorean Triples

The Revelation: Some sets of numbers are "Special Friends." They are whole numbers that perfectly fit the Law without any messy decimals. We call these Pythagorean Triples.
1. Test the Sets: Circle the triples that follow the Law.
(3, 4, 5)    (5, 6, 7)    (5, 12, 13)
(8, 15, 17)    (10, 10, 20)    (7, 24, 25)
2. The Scale Factor: If (3, 4, 5) is a triple, is (30, 40, 50) a triple? Show your math.
Verdict:

Unit 3 Review: The Linear Code

Recall the Path: Before we leave the Triangle, let us remember the Line.
1. Equation to Line: What is the slope and intercept of $y = 3x - 4$?

$m = $ $b = $

2. Proportional Truth: Write an equation for a line that passes through the Origin with a slope of 5.

Code:

The Prophet's Vision: Part 2

4. The Shepherd's Shortcut: A shepherd is at the southwest corner of a rectangular field that is 30 yards wide and 40 yards long. His sheep are at the northeast corner. If he walks diagonally across the field instead of around the edges, how many yards does he save?
Diagonal distance:
Distance saved:
5. The Tent of Meeting: A tent pole is 12 feet high. A rope is tied to the top of the pole and anchored to the ground 5 feet away from the base. How long is the rope?
Calculation: Rope length:
6. The City Wall: A guard tower is 24 feet high. A ramp leading to the top starts 7 feet away from the base of the tower. How long is the ramp?
Calculation: Ramp length:

Unit 4 Mastery Review: The Plumb Line

The Synthesis: Prove that you are a Master of the Corner.
1. The Unknown Leg: A right triangle has a hypotenuse of 10 and one leg of 6. How long is the other leg? (Hint: $6^2 + b^2 = 10^2$).
Calculation: Leg $b = $
2. The True Square: You measure a corner. One side is 9 inches, the other is 40 inches. The diagonal is 41 inches. Is this a perfect $90^\circ$ corner? Prove it with math.
Proof:
3. The 3D Challenge: A box is 3cm wide, 4cm long, and 12cm high. Find the diagonal across the floor ($3^2+4^2$). Then use that result as a leg to find the diagonal across the entire box!
Floor Diagonal: Box Diagonal:

The Scribe's Journal: Reflections on Unit 4

Meditation: The Invisible Bridge

You have learned that by squaring two visible paths (the legs), you can find the length of an invisible path (the hypotenuse). You are calculating distances you haven't even walked yet.

Reflect: How does it feel to know that truth allows you to "see" things that are hidden? Does the Pythagorean Law make you feel more confident about the structure of the world?

The Scribe's Dictionary: Unit 4 Vocabulary

Pythagorean Theorem:
The law $a^2 + b^2 = c^2$, which connects the three sides of a right triangle.
Right Triangle:
A shape with a perfect square corner ($90$ degrees).
Hypotenuse ($c$):
The longest side of a right triangle, stretching across from the right angle.
Legs ($a$ and $b$):
The two shorter sides that form the right angle.
Square Root ($\sqrt{x}$):
The number that was multiplied by itself to make $x$. The "Seed" of the square.
Theorem:
A mathematical truth that has been proven and cannot be broken.

C.A.M.E. Verification Node: The Builder's Audit

A student calculates $a^2 + b^2 = c^2$ for a triangle with legs of 3 and 4. They write:
Step 1: $3^2 + 4^2 = 9 + 16 = 25$.
Step 2: The hypotenuse is 25.

Is this the Truth? Look at the result. Could a triangle with sides 3 and 4 have a side of 25? What did the student forget to do in Step 2? Find the actual side length.

Verdict:

The Covenant

"I, a Builder of the Master's Kingdom, solemnly promise to respect the law of the triangle. I will check my corners, honor the square, and always use the theorem to find the truth. I know that a firm foundation is built on accurate calculation and honest measurements."

Signed:

Date:

UNIT 4 MASTERED!

You have built a firm foundation.

HALLELUYAH!

📐